# The Stacks Project

## Tag 02PI

Lemma 106.19.6. Let $R$ be a local ring.

1. If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
2. If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \varphi, \psi) = 0$.
3. Suppose that we have a short exact sequence of $2$-periodic complexes $$0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0$$ If two out of three have cohomology modules of finite length so does the third and we have $$e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3).$$

Proof. This follows from Chow Homology, Lemmas 41.2.3 and 41.2.4. $\square$

The code snippet corresponding to this tag is a part of the file obsolete.tex and is located in lines 2310–2336 (see updates for more information).

\begin{lemma}
\label{lemma-periodic-length}
Let $R$ be a local ring.
\begin{enumerate}
\item If $(M, N, \varphi, \psi)$ is a $2$-periodic complex
such that $M$, $N$ have finite length. Then
$e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
\item If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex
such that $M$ has finite length. Then
$e_R(M, \varphi, \psi) = 0$.
\item Suppose that we have a short exact sequence of
$2$-periodic complexes
$$0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0$$
If two out of three have cohomology modules of finite length so does
the third and we have
$$e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3).$$
\end{enumerate}
\end{lemma}

\begin{proof}
This follows from Chow Homology, Lemmas
\ref{chow-lemma-finite-periodic-length}.
\end{proof}

Comment #2608 by Ko Aoki on June 24, 2017 a 6:02 am UTC

Typo in the statement of (3): "$(2,1)$-periodic complexes" should be replaced by "$2$-periodic complexes."

Comment #2631 by Johan (site) on July 7, 2017 a 12:48 pm UTC

Thanks, fixed here.

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